Paradox 2 - the set that cannot contain itself

The set of all sets that do not contain themselves as members cannot contain itself, therefore it both does and doesn't contain itself.

Or to put it more simply:

A popularised version of Bertrand Russell's paradox is to imagine a place where there is a male Barber. He shaves all the men who do not shave themselves.

Does he shave himself?

He cannot, because he can only shave those who do not shave themselves. If he does not, then he does not shave himself and therefore is part of the set of those who do not shave themselves, and he must shave himself...

Wittgenstein claimed to have solved this paradox (Russell's theory of types) in his 'Tractatus'. Essentially he claimed it showed a problem with using language to describe the world and used logical symbols to refute it. You can read more on this here, at project Euclid